A first step is to save trees by cutting down on the number of tissues you use in the first place. (Hanky, anyone?) But back to the question: In theory, throwing your tissues down the toilet—in the course of your regularly scheduled flushes—might have slight benefits: Some paper would dissolve, eventually ending up as part of a sludge that might be recycled into fertilizer or converted into electricity. But given that tissues aren't designed to disintegrate in water, much of the paper may end up getting filtered out during the sewage treatment process. From there, it goes right to the dump, just as it would if you threw it in the trash. (Of course, you may be wasting energy by forcing the sewage treatment plant to handle extra material.) Many municipal sewer systems ask residents not to dump tissues down the toilet to prevent the risk of clogged drains—although the Lantern's understanding is that this rule is designed to keep people from using their pipes as an all-purpose wastebasket.
Ultimately, your choice of where to dispose your snotty tissues is basically a wash. When it comes to the environment, there are weightier things to worry about than tissues.
Is there an environmental quandary that's been keeping you up at night? Send it to firstname.lastname@example.org, and check this space every Tuesday.